> On 9 Jan., 17:01, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:>>> > It would be countable if "countable" was a sensible notion. Everybody>> > who believes that "countable" is a sensible notion and that the power>> > set axiom is valid must think that every subset of a countable set>> > exists and is countable.>>>> Wrong: Brouwer did not think so, for one.>> Brouwer did not think so because he denied uncountability.

That is simply irrelevant to your false claim.

> I said "everybody who believes that the power set axiom is valid".> That is tantamount with uncountably many subsets of |N.

Not according to Brouwer.

>> More importantly, for WMathematics, you make assertions about a>> non-existent set: there is no such set of finitely defined reals.>> There is the set of all finite definitions (and the set of all> languages). A finitely defined real is defined by one or more finite> definitions. And I assume that a subset has not larger cardinality> than its super set.

You are already talking about a non-existent set.

> You may deny that.

*You* deny that there is such a set as the set of all finitely definedreals.

> In standard set theory it is> accepted.

So what? We're talking about WMathematics here, which growsmurkier by the day.